degree of abstraction, for example, approximating a
game with 10165 states by one with 1014 states. There
has been some recent work done on abstraction algorithms with theoretical guarantees, though that work
does not scale to games nearly as large as no-limit
Texas hold ’em. One line of work performs lossless
abstraction, which guarantees that the abstract game
is exactly isomorphic to the original game (Gilpin
and Sandholm 2007b). This work has been applied to
compute equilibrium strategies in Rhode Island hold
’em, a medium-sized ( 3. 1 billion nodes) variant of
poker. Recent work has also presented the first lossy
abstraction algorithms with bounds on the solution
quality (Kroer and Sandholm 2014). However, the
algorithms are based on integer programming formulations, and only scale to a tiny poker game with
a five-card deck. It would be very interesting to bridge
this gap between heuristics that work well in practice
for large games with no theoretical guarantees, and
the approaches with theoretical guarantees that have
more modest scalability.

Scalable algorithms for computing Nash equilibria
have diverse applications, including cybersecurity
(for example, determining optimal thresholds to protect against phishing attacks), business (for example,
auctions and negotiations), national security (for
example, computing strategies for officers to protect
airports), and medicine. For medicine, algorithms
that were created in the course of research on
poker (Johanson et al. 2012) have been applied to
compute robust policies for diabetes management
(Chen and Bowling 2012); recently it has been proposed that equilibrium-finding algorithms are applicable to the problem of treating diseases such as the
HIV virus that can mutate adversarially (Sandholm

2015).

For the pseudoharmonic action translation mapping, in addition to showing that it outperforms the
best prior approach in terms of exploitability in several games, we have also presented several axioms
and theoretical properties that it satisfies; for example, it is Lipschitz continuous in A and B, and therefore robust to small changes in the actions used in
the action abstraction (Ganzfried and Sandholm
2013). Another mapping that has very high
exploitability in several games also satisfies these
axioms, and further investigation can lead to deeper
theoretical understanding of this problem and potentially new improved approaches.

Even the postprocessing approaches, which appear
to be purely heuristic, have interesting theoretical
open questions. For example, it has been shown that
purification (that is, selecting the highest-probability
action with probability 1) leads to an improved performance in uniform random 4 x 4 matrix games
using random 3 x 3 abstractions when playing
against the Nash equilibrium of the full 4 x 4 game
for the opponent (Ganzfried, Sandholm, and Waugh

2012). These results were based off simulations thatwere statistically significant at the 95 percent confi-dence level, and it would be interesting to provide aformal proof. Furthermore, that paper provided aconjecture for the specific supports of the games forwhich the approach would improve or not changeperformance, which was also based on statistically-significant simulations. It would be interesting toprove this formally as well, and to generalize theresults to games of arbitrary size. On a broader level,there is relatively little theoretical understanding forwhy the postprocessing approaches — which onewould expect to make the strategies more predictable— have been shown to be consistently successful.Surprisingly, the improvements in empirical per-formance do not necessarily come at the expense ofworst-case exploitability, and a degree of threshold-ing has been demonstrated to actually reduceexploitability for a limit Texas hold ’em agent(Ganzfried, Sandholm, and Waugh 2012).

AcknowledgementsThe competition was organized by Professor TuomasSandholm, and the agent was created by NoamBrown, Sam Ganzfried, and Tuomas Sandholm. Someof the work described was performed while theauthor was a student at Carnegie Mellon Universitybefore the completion of his Ph.D.; the article reflectsthe views of the author alone and not necessarilythose of Carnegie Mellon University. The work doneat Carnegie Mellon University was supported by theNational Science Foundation under grants IIS-1320620, IIS-0964579, and CCF-1101668, as well asXSEDE computing resources provided by the Pitts-burgh Supercomputing Center.

Note

1. See the Gurobi Optimizer Reference Manual Version
6.0 available from Gurobi Optimization, Inc.
( www.gurobi.com).